nLab irreducible element

Contents

Context

Algebra

Monoid theory

Contents

Definition

In a monoid, an element xx is irreducible if it is neither invertible nor the product of two non-invertible elements. Without bias, we can say that xx is irreducible if, whenever it is written as a product of a finite list of elements, all but one elements in the list are invertible.

In a commutative ring, an element is irreducible if it is neither invertible nor the product of two non-invertible elements, with respect to the multiplication operation on the commutative ring.

Examples

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